So first tip:
cat /proc/sys/kernel/printk_ratelimit shows you the amount of seconds the ratelimiter suppresses messages. The default is 5 seconds and you can adjust it to more reasonable values in case you get heavily flooded like in the example above. Notice that this will mean your dmesg becomes rather useless as the kernel is not very selective about which messages to suppress.

Now when you google "Neighbour table overflow", you'll find thousands of pages suggesting to increase the arp / lladdr caches and garbage collection (gc) times like so:

That hints at what is really happening here: The bridge confused the link-local router negotiation and so you get endless ff02:: neighbour routing entries added to your caches until they flow over. So increasing the caches as in the sysctl entries above is basically pasting band-aid over the problem.

ip route show cache table all will show you the tables. With all entries. See if you have too many ff02:: neighbours in there.
If so, you should try to add change your /etc/network/interfaces on Debian / Ubuntu similar to this:

Red Hat/CentOS users will need to adjust the config spread throughout multiple files in /etc/sysconfig/network-scripts.
The ifup-ipv6 script is a good one to look at and amend.

The increase of the hash_max entry makes your bridge survive the initial storm of (useless) router solicitations.
multicast_snooping is usually off when routing but you may need it to make sure your VMs on the bridge can be reached.
Finally we make sure the bridge does not accept router announcements. Because that is what the host system should handle.

Sometimes you may need to throw in a static route or two to reach the VMs. P&P, you remember ... ip -6 neigh add nud permanent proxy <VM:IPv6:goes::here> dev vmbr0 is your friend. Unfortunately the antidote for the dreaded "Neighbour table overflow" depends on the specific cause. So you'll have to poke around a bit. tcpdump -i eth0 -v ip6 will show you what is on the wire and tcpdump -i vmbr0 -v ip6 what's visible on the bridge.